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Related Experiment Video

Updated: Jan 2, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Disentangling orthogonal matrices.

Teng Zhang¹, Amit Singer²

¹Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA.

Linear Algebra and Its Applications

|January 16, 2018

Summary

This summary is machine-generated.

We developed a new algorithm for solving linear systems with orthogonal matrices, inspired by cryo-electron microscopy. This method, using semi-definite programming, outperforms direct solving for molecular reconstruction tasks.

Keywords:

15A24 90C22 Cryo-EM Orthogonal Procrustes problem SDP relaxation

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Last Updated: Jan 2, 2026

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Published on: August 30, 2013

Area of Science:

Computational Biology
Applied Mathematics
Structural Biology

Background:

Cryo-electron microscopy (cryo-EM) relies on accurate molecular reconstruction.
Solving linear systems with orthogonal matrices is crucial in various scientific domains.
The orthogonal Procrustes problem is a related, well-studied problem.

Purpose of the Study:

To address the challenge of solving linear systems with two unknown orthogonal matrices.
To generalize existing methods for molecular reconstruction in cryo-EM.
To introduce a novel algorithm with theoretical performance guarantees.

Main Methods:

Development of a semi-definite programming (SDP) relaxation algorithm.
Theoretical analysis of the algorithm's performance guarantees.
Empirical validation and comparison with existing methods.

Main Results:

The proposed SDP relaxation algorithm effectively solves linear systems with two unknown orthogonal matrices.
The algorithm demonstrates superior performance compared to direct solving methods lacking orthogonal constraints.
Theoretical guarantees support the algorithm's efficacy.

Conclusions:

The novel SDP-based algorithm offers a significant advancement for molecular reconstruction in cryo-EM.
This approach provides a robust solution for problems involving orthogonal matrices.
The methodology is extendable to systems with more than two unknown orthogonal matrices.